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Theorem dfiun3 5369
 Description: Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfiun3.1 𝐵 ∈ V
Assertion
Ref Expression
dfiun3 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)

Proof of Theorem dfiun3
StepHypRef Expression
1 dfiun3g 5367 . 2 (∀𝑥𝐴 𝐵 ∈ V → 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵))
2 dfiun3.1 . . 3 𝐵 ∈ V
32a1i 11 . 2 (𝑥𝐴𝐵 ∈ V)
41, 3mprg 2923 1 𝑥𝐴 𝐵 = ran (𝑥𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988  Vcvv 3195  ∪ cuni 4427  ∪ ciun 4511   ↦ cmpt 4720  ran crn 5105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-cnv 5112  df-dm 5114  df-rn 5115 This theorem is referenced by:  tgrest  20944  comppfsc  21316  sigapildsys  30199  ldgenpisyslem1  30200  dstfrvunirn  30510  mblfinlem2  33418  volsupnfl  33425  istotbnd3  33541  sstotbnd  33545  fourierdlem80  40166
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